Question

In the given figure, a square OABC has been inscribed in the quadrant OPBQ. If OA = 20 cm, then the area of the shaded region is


(a) 214 cm²
(b) 228 cm²
(c) 242 cm²
(d) 248 cm²

Answer 1

(b) 228 cm²
Join OB.
Now, OB is the radius of the circle.
$$\begin{gathered} \mathrm{We}\mathrm{have}: \\ O{B}^2=O{A}^2+A{B}^2\left[\mathrm{By}\mathrm{Pythagoras}\text{'}\mathrm{theorem}\right] \\ \Rightarrow O{B}^2=\left\{{\left(20\right)}^2+{\left(20\right)}^2\right\}{\mathrm{cm}}^2 \end{gathered}$$

$$\begin{gathered} \Rightarrow O{B}^2=\left(400+400\right){\mathrm{cm}}^2 \\ \Rightarrow O{B}^2=800{\mathrm{cm}}^2 \\ \Rightarrow OB=20\sqrt{2}\mathrm{cm} \end{gathered}$$

Hence, the radius of the circle is $20\sqrt{2}\mathrm{cm}$.
Now,
Area of the shaded region = Area of the quadrant $-$ Area of the square OABC
$$\begin{gathered} =\left|\left(\frac{1}{4}\times 3.14\times 20\sqrt{2}\times 20\sqrt{2}\right)-\left(20\times 20\right)\right|{\mathrm{cm}}^2 \\ =\left|\left(\frac{1}{4}\times \frac{314}{100}\times 800\right)-400\right|{\mathrm{cm}}^2 \\ =\left(628-400\right){\mathrm{cm}}^2 \\ =228{\mathrm{cm}}^2 \end{gathered}$$